1. Um ponto de vista simbólico sobre a Simulação de Algoritmos Quânticos António Pereira & Rosália Rodrigues CEOC-UA – CIMA-UE 2006

2. CEOC – CIMA - 2006 2 Quantum Computation Research in Quantum Computation: building quantum devices designing algorithms for quantum devices How to Simulate it ?

3. CEOC – CIMA - 2006 3 Simulating Quantum Computation Vectorial approach: state  vector on a Hilbert space evolution  matrix products advantage advantage: easy to implement and trace drawback drawback: exponential growth in space and time Symbolic approach: state  linear expression evolution  algebraic rules advantage advantage: control over complexity drawback drawback: “convince” Mathematica not to evaluate... yet S ymbolic Q uantum C omputer S imulator

4. CEOC – CIMA - 2006 4 qudits & qubits

5. CEOC – CIMA - 2006 5 kets in SQCS Basis qudit state Linear expression of ket objects Object with head ket General qudit state

6. CEOC – CIMA - 2006 6 bras in SQCS Riesz Theorem:

7. CEOC – CIMA - 2006 7 braKets in SQCS conjugate linear in the first argument linear in the second braKet

8. CEOC – CIMA - 2006 8 Qudit Systems ………… 123 n

9. CEOC – CIMA - 2006 9 The Kronecker product in SQCS Properties of the Tensor Product (Kronecker Product): Associative Noncommutative Distributive with respect to linear combinations

10. CEOC – CIMA - 2006 10 Operators in SQCS Quantum Algorithm Initial state + Sequence of unitary operators + Measurement op[name_,n_,f_] Every linear operator is represented in SQCS by an object where: name ― label for the operator n ― number of qudits on which the operator acts f ― function that defines the action of the operator on the basis qudits states (set of rules) The discrete time evolution of a closed quantum system is described by the action of a unitary operator

11. CEOC – CIMA - 2006 11 Operators in SQCS The Hadamard operator Creates a uniform superposition Is its own inverse

12. CEOC – CIMA - 2006 12 Operators in SQCS The Walsh-Hadamard operator

13. CEOC – CIMA - 2006 13 Operators in SQCS The Outer Product operator Completeness Relation:

14. CEOC – CIMA - 2006 14 Simulating Grover’s Algorithm

15. CEOC – CIMA - 2006 15 Simulating Grover’s Algorithm Classical Database Case: Classical Database Quantum Computer f(x)

16. CEOC – CIMA - 2006 16 Simulating Grover’s Algorithm Quantum Database Case: Quantum Database Quantum Computer

17. CEOC – CIMA - 2006 17 Simulating Grover’s Algorithm Step by step: Database of size 2 5 =32 Index of the element to be searched for Number of steps The Oracle Grover’s operator

18. CEOC – CIMA - 2006 18 Simulating Grover’s Algorithm Step by step:

19. CEOC – CIMA - 2006 19 Simulating Grover’s Algorithm Probability distribution Step by step:

20. CEOC – CIMA - 2006 20 Probability distribution Simulating Grover’s Algorithm Step by step:

21. CEOC – CIMA - 2006 21 Probability distribution Simulating Grover’s Algorithm Step by step:

22. CEOC – CIMA - 2006 22 Probability distribution Simulating Grover’s Algorithm Step by step:

23. CEOC – CIMA - 2006 23 Grover’s Algorithm – Simulation Times Grover’s Algorithm – Simulation Times Time × Database size Time × Number of qubits Classical Database Case: Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM

24. CEOC – CIMA - 2006 24 Grover’s Algorithm – Simulation Times Grover’s Algorithm – Simulation Times Time × Database size Time × Number of qubits Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM Quantum Database Case:

25. CEOC – CIMA - 2006 25 Conclusions & Further work Measuring Operators. A quantum register address manager. Simulate other quantum algorithms: Deutsch-Jozsa, Shor, … Use SQCS as a tool for the development of new quantum algorithms. Symbolic Approach to Quantum Computation: Provides a suitable environment for testing quantum algorithms. Allows for larger problem instances. Algorithms can be programmed at high-level. Useful tool for the teaching of Quantum Computation. Conclusions: Further work:

26. CEOC – CIMA - 2006 26 1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000) 2. Kitaev, A.Y., Shen, A., Vyalyi, M.: Classical and quantum computation. Volume 47 of Graduate Studies in Mathematics. American Mathematical Society (2002) 3. Wolfram, S.: The Mathematica Book, Fifth Edition. Wolfram Media, Inc. (2003) 4. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. 28th Annual ACM Symposium on the Theory of Computing. (1996) 212-219 5. Biham, E., Biham, O., Biron, D., Grassl, M., Lidar, D.A.: Grover's quantum search algorithm for an arbitrary initial amplitude distribution. Physical Review A 60 (1999) 27-42 6. Pereira, António, Rodrigues, Rosália: A Symbolic Approach to Quantum Computation Simulation. Lecture Notes in Computer Science (2006) Vol. 3992. 454 – 461 7. Pereira, António, Rodrigues, Rosália: Symbolic Quantum Computation Simulation with Mathematica. Cadernos de Matemática. Universidade de Aveiro. CM05/I-44 (2005) References Thank You